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Let's solve the problems one by one.

### 1. Find the inverse of \( f(x) = x^5 \)
To find the inverse of this function:

- Start by writing \( y = x^5 \).
- Solve for \( x \) in terms of \( y \):
  \[
  y = x^5 \implies x = \sqrt[5]{y}
  \]
- Now, switch \( y \) and \( x \) to express the inverse:
  \[
  f^{-1}(x) = \sqrt[5]{x}
  \]
So, the inverse is \( f^{-1}(x) = \sqrt[5]{x} \).

### 2. Find the inverse of \( f(x) = \sqrt[9]{x + 3} \)
For this one:

- Write \( y = \sqrt[9]{x + 3} \).
- Eliminate the radical by raising both sides to the power of 9:
  \[
  y^9 = x + 3
  \]
- Solve for \( x \):
  \[
  x = y^9 - 3
  \]
- Finally, switch \( y \) and \( x \):
  \[
  f^{-1}(x) = x^9 - 3
  \]
So, the inverse is \( f^{-1}(x) = x^9 - 3 \), which matches the provided answer.

### 3. Find the inverse of \( f(x) = (x + 1)^7 - 3 \)
For this function:

- Write \( y = (x + 1)^7 - 3 \).
- Solve for \( x \). First, add 3 to both sides:
  \[
  y + 3 = (x + 1)^7
  \]
- Take the 7th root of both sides:
  \[
  \sqrt[7]{y + 3} = x + 1
  \]
- Subtract 1 from both sides:
  \[
  x = \sqrt[7]{y + 3} - 1
  \]
- Switch \( y \) and \( x \):
  \[
  f^{-1}(x) = \sqrt[7]{x + 3} - 1
  \]
So, the inverse is \( f^{-1}(x) = \sqrt[7]{x + 3} - 1 \).

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Here are 5 related questions:
1. How do you verify if two functions are inverses of each other?
2. What are the conditions for a function to have an inverse?
3. How would you find the inverse of a logarithmic function?
4. Can all polynomial functions be inverted? Why or why not?
5. How does the graph of a function compare to the graph of its inverse?

**Tip:** Remember, when solving for the inverse of a function, the goal is to express the dependent variable in terms of the independent variable, then swap them!

1. Find the inverse of f(x) = x^5. 2. Find the inverse of f(x) = 9th root of (x + 3). 3. Find the inverse of f(x) = (x + 1)^7 - 3.

Learn how to find the inverse of functions such as f(x) = x^5, f(x) = 9th root of (x + 3), and f(x) = (x + 1)^7 - 3. This step-by-step solution explains how to handle inverse functions with polynomials and radicals.

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